Abel - I don't have a lot of experience with either caliber, but I am in the process of selecting and purchasing a medium caliber rifle, and both of those calibers are on my short list of possible/acceptable selections. I'll tell you why I might choose the .243 vs. the .25-06.
I am currently, and heavily, leaning toward the .243. I reload, but one of my key criteria for this rifle is easy, and common, commercial ammunition availability. (The .308 keeps popping its head up because of this issue, but I keep whacking it back down with my Whack-a-Mole bat - it's still on the list, but what I really want is a light-kicking rifle that doesn't wear me out to shoot.) I see .243 just about everywhere. I don't see .25-06 as often, and when I do, it costs more than .243. This issue has knocked the very nice .260 Remington off my list completely, at least for this rifle, even though I have been interested in it since its introduction.
I am buying this rifle to *shoot*, and I intend to shoot the heck out of it. I don't know exactly how long a hunting rifle barrel will last, but I intend to find out. I will hunt coyotes, deer, groundhogs, maybe feral hogs, and probably some pdogs at least once with this rifle. (It may sound crazy to some, but I may even stretch it one day to elk. I, myself, have only used a .30'06 on elk, but I know a number of people who use the .25-06, 6mm Remington, and .243, and are successful doing so.) I will also target shoot regularly at my club. This is where my new rifle will get most of its use, and probably where the original barrel will die. It will be used on paper and steel out past 600 yards on our long rifle range. I know that there are target and match bullets available for the .25, but there are a lot more choices available for the 6mm/.243 because there are so many 6mm cartridges used for long range match shooting. (.308 just got whacked in the head again.)
There are a lot of similar-performing cartridges out there. Every rifleman has to select his rifle based on his own priorities. I don't think that the .243 has huge advantages over the .25-06, even though I will probably buy one over the .25. What I do think is that they are both excellent cartridges that do about the same thing with bullets of similar size and weight. The .25 can push them just a little bit faster, but ... really, at the speeds they are going, I can't see that as being a huge advantage. I'm not going to be shooting at big game far enough for that to matter, and at the range or on pdogs, it will be a matter of practicing and calculating bullet drop, for which actual velocity is only one part of the equation.
I don't mind long actions, but I am currently favoring short actions because the rifles tend to be a bit shorter and more compact, and weigh a bit less - when I hunt, I hike a lot, and I would prefer to save a bit of weight if all else is equal. This, again, favors the .308-based cartridge, rather than the .30'06-based cartridge.
Some of the rifles I am looking at are not available in the .25-06 because it is not as common a chambering as some others. It's kind of a fun process, shopping for a rifle. You have to decide what's most important - the rifle, or the cartridge. Then, you start looking at either rifles that are available in your chosen chambering, or at the list of available chamberings for the rifle you are interested in.
In the end, and probably within the next couple of weeks, I'll buy a rifle. The equation will include caliber, price, features, and a few other things, and I have a short list of what I think I'm going to select my rifle from, but if someone offers me a rifle I really like at a price I can't ignore, I'll just check to make sure the caliber is one of the ones I figure is 'okay' for my purposes, and buy a new set of dies to go with whatever it is. The chambering, as long as it's one of the ones on my list, will be just fine, whatever it is.